Mesh Basics Mesh Basics 1 Spring 2010
Definitions: Definitions: 1/2 Definitions: Definitions: 1/2 1/2 1/2 � A polygonal mesh consists of three kinds of mesh elements : vertices, edges, and faces. � The information describing the mesh elements are mesh connectivity and mesh geometry . y g y � The mesh connectivity , or topology, describes the incidence relations among mesh elements the incidence relations among mesh elements ( e.g ., adjacent vertices and edges of a face, etc). � The mesh geometry specifies the position and � The mesh geometry specifies the position and other geometric characteristics of each vertex. 2
Definitions: Definitions: 2/2 Definitions: Definitions: 2/2 2/2 2/2 closed fan � A mesh is a manifold if (1) (1) each edge is incident to only one or two faces and (2) (2) the faces incident to a vertex form a closed or an open fan. form a closed or an open fan. � The orientation of a face is a cyclic open fan order of the incident vertices. � The orientation of two adjacent faces is compatible , if the two vertices of the common edge are in opposite the common edge are in opposite order. � A manifold mesh is orientable if any y two adjacent faces have compatible orientation. 3
Non Non Manifold Non Non-Manifold Manifold Meshes Manifold Meshes Meshes Meshes � Manifold Conditions: (1) (1) Each edge is incident to only one or two faces and (2) (2) the faces incident to a vertex form a closed or an open fan. � The following examples are not manifold meshes! g p 4
Manifolds Manifolds w ith/w ithout Manifolds Manifolds w ith/w ithout w ith/w ithout Boundary ith/w ithout Boundary Boundary Boundary � If every vertex has a closed fan, the given manifold has no boundary. Edges only incident to one face form the boundary of the manifold. � Boundary is a union of simple polygons. y p p yg � We only consider We only consider orientable orientable manifolds manifolds w i w ithout t out out bou out bou bou da bounda da y dary in in thi t his cou s cou course cou se se. se manifold boundary closed fan open fan 5 boundary non-manifold boundary
Non Non-Orientable Non Orientable Non Orientable Manifolds: Orientable Manifolds: Manifolds: 1/2 anifolds: 1/2 1/2 1/2 � Not all manifolds are orientable. The most well- known ones are Möbius band and Klein bottle. � The Möbius band is shown below, and is an one- sided manifold with boundary ( i.e ., a circle). y http://www.jcu.edu/math/vignettes/Mobius.htm 6
Non Non Orientable Non Non-Orientable Orientable Manifolds: Orientable Manifolds: Manifolds: 2/2 anifolds: 2/2 2/2 2/2 � The Klein bottle is a manifold without boundary. � Slicing a Klein bottle properly yields two Möbius g p p y y bands. A Klein bottle sliced to show its interior A Klein bottle sliced to show its interior. However, Klein bottles have no interior. Maurice Fréchet and Ky Fan, Invitation to Combinatorial Topology p gy 7 Tin-Tin Yu, MTU
Mesh Mesh Examples: Mesh Mesh Examples: Examples: 1/2 Examples: 1/2 1/2 1/2 � Vertices: 4,634 � Edges: 13,872 � Faces:9,248 8
Mesh Mesh Examples: Mesh Mesh Examples: Examples: 2/2 Examples: 2/2 2/2 2/2 � Vertices: 703 � Edges: 2106 � Faces: 1401 9
Euler Euler Poincar Euler Euler-Poincar Poincaré Characteristic: Poincaré Characteristic: Characteristic: 1/5 haracteristic: 1/5 1/5 1/5 � Given a 2-manifold mesh M without boundary, the Euler-Poincaré characteristic of M is χ ( M ) = V-E+F, where V, E and F are the number of vertices, number of edges, and number of faces. V=8 E=12 F=6 V=8, E=12, F=6 V=16 E=32 F=16 V=16, E=32, F=16 V=28 E=56 F=26 V=28, E=56, F=26 χ ( M ) =V-E+F=2 χ ( M ) =V-E+F=0 χ ( M ) =V-E+F=-2 10
Euler Euler Poincar Euler Euler-Poincar Poincaré Characteristic: Poincaré Characteristic: Characteristic: 2/5 haracteristic: 2/5 2/5 2/5 � Euler-Poincaré characteristic χ ( M ) = V-E+F is independent of tessellation. V=24, E=48, F=22 χ ( M ) =V-E+F=-2 V=16, E=32, F=16 V 16, E 32, F 16 V=28, E=56, F=26 V 28, E 56, F 26 V=16 E=36 F=20 V 16, E 36, F 20 χ ( M ) =V-E+F=0 χ ( M ) =V-E+F=-2 χ ( M ) =V-E+F=0 11
Euler Euler Poincar Euler Euler-Poincar Poincaré Characteristic: Poincaré Characteristic: Characteristic: 3/5 haracteristic: 3/5 3/5 3/5 � An orientable 2-manifold mesh M M with g “ handles handles ” ( i.e ., genus genus ) has Euler-Poincaré characteristic χ ( M ) = V-E+F = 2(1- g ). � Spheres, boxes, tetrahedrons and convex p surfaces have g = 0; but, tori have g = 1. g =0 ⇒ χ ( M ) =2(1-0)=2 g =1 ⇒ χ ( M ) =2(1-1)=0 g =2 ⇒ χ ( M ) =2(1-2)=-2 12
Euler Euler Poincar Euler Euler-Poincar Poincaré Characteristic: Poincaré Characteristic: Characteristic: 4/5 haracteristic: 4/5 4/5 4/5 � The boundary of an orientable 2-manifold is the union of a set of simple polygons. � Since each polygon bounds a face, these “boundary faces” may be added back to form a y y manifold without boundary so that Euler- Poincaré characteristic can be applied. � The Euler-Poincaré characteristic of an orientable 2-manifold with boundary is χ ( M ) = o e b e o d w bou d y s χ ( ) 2(1- g )- ∂ , where ∂ is the number “boundary polygons”. po ygo s . 13
Euler Euler Euler-Poincar Euler Poincar Poincaré Characteristic: Poincaré Characteristic: Characteristic: 5/5 haracteristic: 5/5 5/5 5/5 � Two Examples: V = 10, E = 15, F = 6 V = 30, E = 54, F = 20 g 0, ∂ g = 0, ∂ = 1 g = 2, ∂ = 2 g 2, ∂ 1 2 χ ( M ) = V-E+F = 1 χ ( M ) = V-E+F = -4 14 χ ( M ) = 2(1- g )- ∂ = 1 χ ( M ) = 2(1- g )- ∂ = -4
Homeomorphisms: Homeomorphisms: 1/3 Homeomorphisms: Homeomorphisms: 1/3 1/3 1/3 � Two 2-manifold meshes A and B are homeomorphic if their surfaces can be homeomorphic transformed to the other by twisting, squeezing, and stretching without cutting and gluing. � Thus, boxes, spheres and ellipsoids are homeomorphic to each other. is homeomorphic to 15
Homeomorphisms: Homeomorphisms: 2/3 Homeomorphisms: Homeomorphisms: 2/3 2/3 2/3 � Two orientable 2-manifold meshes without boundary are homeomorphic if and only if they have the same Euler-Poincaré characteristic. � Thus, a m -handle ( i.e ., genus m ) orientable g mesh is homeomorphic to a n -handle ( i.e ., genus n ) orientable mesh if and only if m = n . � Two orientable 2-manifold meshes with the same number of boundary polygons are same number of boundary polygons are homeomorphic if and only if they have the same Euler-Poincaré characteristic. u e o ca é c a acte st c. 16
Homeomorphisms: Homeomorphisms: 3/3 Homeomorphisms: Homeomorphisms: 3/3 3/3 3/3 � Hence, any orientable 2-manifold mesh without boundary is homeomorphic to a sphere with m handles ( i.e ., genus m ), where m ≥ 0. 17
Applications: Applications: 1/3 Applications: Applications: 1/3 1/3 1/3 � A mesh is regular if all faces have the same number edges, and all vertices are incident to the same number of edges ( i.e ., valence). � Each face of a regular quad mesh is a g q quadrilateral ( i.e ., four-sided) and each vertex is incident to four edges ( i.e ., valence = 4). 18
Applications: Applications: 2/3 Applications: Applications: 2/3 2/3 2/3 � Only a torus can be a regular quad mesh! � Since each vertex has 4 edges and each edge is g g counted twice, we have 4V = 2E ( i.e ., V=E/2). � Since each face has 4 edges and each edge is � Since each face has 4 edges and each edge is counted twice, we have 4F = 2E ( i.e ., F = E/2). � Thus χ ( M ) = V-E+F = 0 means a torus! � Thus, χ ( M ) = V-E+F = 0 means a torus! 19
Applications: Applications: 3/3 Applications: Applications: 3/3 3/3 3/3 � Only tori can be regular triangle mesh of valence 6! � Since each vertex has 6 edges and each is counted g twice, we have 6V = 2E ( i.e ., V=E/3). � Since each face has 3 edges and each edge is � Since each face has 3 edges and each edge is counted twice, we have 3F = 2E ( i.e ., F = 2E/3). � Thus χ ( M ) = V-E+F = 0 means a torus! � Thus, χ ( M ) = V-E+F = 0 means a torus! 20
Data Data Structures Data Data Structures Structures for Structures for for Meshes for Meshes Meshes eshes � Since meshes are usually large and complex and since many operations are performed on meshes, compact data structures that support efficient algorithms are needed. � Depending on the applications in hand, one may use vertex- (or point-) based, edge-based, face-based, or other data structures. � One of the earliest edge-based data structure is � One of the earliest edge based data structure is the winged-edge data structure. Its new variant is the half-edge data structure. s t e half edge data st uctu e. 21
What What What Is What Is Is a Winged Is a Winged Winged-Edge? inged Edge? Edge? 1/7 Edge? 1/7 1/7 1/7 � If all faces are oriented clock-wise, each edge has eight pieces of incident information. • Given edge: b = XY g a a d d Y Y • Incident faces: 1 and 2 • Pred. & succ. edges of 1 • Pred. & succ. edges of 2 P d & d f 2 b 1 2 • The wings of edge b = XY • The wings of edge b = XY are faces 1 and 2. • Edge b is a winged-edge g g g c e X 22
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